Scenario reduction algorithms can be an effective means to provide a tractable description of the uncertainty in optimal control problems. However, they might significantly compromise the performance of the controlled system. In this paper, we propose a method to compensate for the effect of scenario reduction on stochastic optimal control problems for chance-constrained linear systems with additive uncertainty. We consider a setting in which the uncertainty has a discrete distribution, where the number of possible realizations is large. We then propose a reduction algorithm with a problem-dependent loss function, and we define sufficient conditions on the stochastic optimal control problem to ensure out-of-sample guarantees (i.e., against the original distribution of the uncertainty) for the controlled system in terms of performance and chance constraint satisfaction. Finally, we demonstrate the effectiveness of the approach on a numerical example.

We present a model predictive control framework for a class of nonlinear systems affected by additive stochastic disturbances with (possibly) unbounded support. We consider hard input constraints and chance state constraints and we employ the unscented transform method to propagate the disturbances over the nonlinear dynamics in a computationally efficient manner. The main contribution of our work is the establishment of sufficient conditions for stability and recursive feasibility of the closed-loop system, based on the design of a terminal cost and a terminal set. We focus here on a special class of nonlinear systems that exhibit contractive properties in the dynamics. By assuming this property, we propose a novel approach to efficiently compute the terminal conditions without the need of performing any linearization of the dynamics. Finally, we provide an illustrative example to corroborate our theoretical findings.